Q. positive divided by a negative equals

Q: What is a positive divided by a negative?

It is negative. In general, for a,b>0: \frac{+a}{-b} = -\frac{a}{b} .

Q: Why does a positive divided by a negative give a negative?

Division is inverse of multiplication. If x=\frac{+a}{-b}, then x\cdot(-b)=+a; so x must be negative to make a positive product.

Q: Give a concrete example.

Example: \frac{6}{-2} = -3 because (-3)\cdot(-2)=6.

Q: What about zero divided by a negative?

Zero divided by any nonzero number is zero: \frac{0}{-5}=0 .

Q: Can you divide by zero?

No. Division by zero is undefined: \frac{a}{0} has no meaning for any nonzero a.

Q: How do you handle a negative in the denominator?

Move the sign to the numerator: \frac{a}{-b} = -\frac{a}{b} . Also \frac{-a}{-b}=\frac{a}{b} .

Q: What if both numbers are negative?

Negative divided by negative is positive: \frac{-6}{-2}=3 .

Answer

Let \(x>0\) and \(y>0\). Consider \( -x + y \).

– If \(y > x\): \( -x + y = y – x > 0\) (positive).
– If \(y = x\): \( -x + y = 0\).
– If \(y < x\): \( -x + y = -(x - y) < 0\) (negative).

Detailed Explanation

Explanation

State what “positive divided by a negative” means.

Let \( a \) be a positive number and \( b \) be a negative number, so that \( a > 0 \) and \( b < 0 \).
Rewrite division as multiplication by the reciprocal.

Division can be written as multiplication by the reciprocal. Thus:

\[ \frac{a}{b} = a \cdot \frac{1}{b} \]
Determine the sign of the reciprocal.

The reciprocal of a negative number is also negative. Since \( b \) is negative, \( \frac{1}{b} \) is negative.
Multiply a positive by a negative.

A positive number multiplied by a negative number gives a negative result. Therefore:

\[ a \cdot \frac{1}{b} \text{ is negative.} \]
Conclude the sign of the quotient.

Combining the previous steps,

\[ \frac{a}{b} \text{ is negative.} \]

In words: a positive number divided by a negative number equals a negative number.
Example for clarity.

Take \( a = 6 \) and \( b = -2 \). Then:

\[ \frac{6}{-2} = 6 \cdot \frac{1}{-2} = 6 \cdot \left(-\tfrac{1}{2}\right) = -3. \]

This confirms the rule: positive divided by negative is negative.

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FAQs

What is a positive divided by a negative?

It is negative. In general, for a,b>0: \( \frac{+a}{-b} = -\frac{a}{b} \).

Why does a positive divided by a negative give a negative?

Division is inverse of multiplication. If \(x=\frac{+a}{-b}\), then \(x\cdot(-b)=+a\); so x must be negative to make a positive product.

Give a concrete example.

Example: \( \frac{6}{-2} = -3 \) because \((-3)\cdot(-2)=6\).

What about zero divided by a negative?

Zero divided by any nonzero number is zero: \( \frac{0}{-5}=0 \).

Can you divide by zero?

No. Division by zero is undefined: \( \frac{a}{0} \) has no meaning for any nonzero a.

Is there a simple sign rule for division and multiplication?

Yes: same signs give a positive result; different signs give a negative result.

How do you handle a negative in the denominator?

Move the sign to the numerator: \( \frac{a}{-b} = -\frac{a}{b} \). Also \( \frac{-a}{-b}=\frac{a}{b} \).

What if both numbers are negative?

Negative divided by negative is positive: \( \frac{-6}{-2}=3 \).

Do decimals and fractions follow the same sign rules?

Yes. Perform the numeric division (decimal or fraction) and then apply the sign rule: same → positive, different → negative.

Positive ÷ negative gives a negative.
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